Prace Naukowe Uniwersytetu Śląskiego nr 2769, Katowice
WELL-POSEDNESS OF THE FIXED POINT PROBLEM FOR CERTAIN ASYMPTOTICALLY REGULAR MAPPINGS
Mohamed Akkouchi
Abstract. We study the well-posedness of the fixed point problem for asymp- totically regular self-mappings of a complete metric space (X, d) which satisfy the contractive condition (2.1) described below. This contractive condition is a variant of the contractive condition considered in [6]. The results of this paper provide some improvements and extensions to the results of Ćirić [6], Sharma and Yuel [19], and Guay and Singh [7]. This work is inspired and motivated by the paper [6].
1. Introduction
The notion of well-posednes of a fixed point problem has generated much interest to a several mathematicians, for examples, F.S. De Blassi and J.
Myjak (see [2]), S. Reich and A. J. Zaslavski (see [16]), B.K. Lahiri and P.
Das (see [9]) and V. Popa (see [14] and [15]).
Definition 1.1. Let (X, d) be a metric space and T : (X, d) → (X, d) a mapping. The fixed point problem of T is said to be well posed if:
(a) T has a unique fixed point z in X;
(b) for any sequence {xn} of points in X such that limn→∞d(T xn, xn) = 0, we have limn→∞d(xn, z) = 0.
Received: 15.06.2009. Revised: 05.08.2009.
(2010) Mathematics Subject Classification: 54H25, 47H10.
Key words and phrases: well-posedness, fixed point problem, fixed points, self- mappings, asymptotically regular mappings, complete and non complete metric spaces, orbitally complete spaces.
The notion of contractive mapping has been introduced by Banach in [1].
In the last thirty years there have appeared different types of generalizations of this concept. The connection between them has been studied in different papers (see [8], [10], [17] and [20]).
We recall that the idea of implicit relation is due to V. Popa (see [11]
and [12]). Since the publication of the papers [11] and [12], almost all the contractive conditions involved in fixed or common fixed point results are defined by implicit relations.
Browder and Petryshyn (see [3]) defined the following notion.
Definition 1.2. A selfmapping T on a metric space (X, d) is said to be asymptotically regular at a point x in X, if
(1.1) lim
n→∞d(Tnx, TnT x) = 0, where Tnx denotes the n-th iterate of T at x.
Almost all of the contractive conditions ensuring the existence of fixed points and generalizing the Banach principle imply the asymptotic regularity of the mappings under consideration. So the investigation of the asymptoti- cally regular maps plays an important role in fixed point theory.
Ćirić (see [6]) pointed out that Sharma and Yuel [19] and Guay and Singh [7] were among the first who used the concept of asymptotic regularity to prove fixed point theorems.
In [6], Ćirić generalized the results of Sharma and Yuel [19] and Guay and Singh [7] and studied a wide class of asymptotically regular mappings which possess fixed points in complete metric spaces and has proved the following theorem.
Theorem 1.1. Let R+ be the set of nonnegative reals and let Fi: R+ → R+ be functions such that Fi(0) = 0 and Fi is continuous at 0 for i = 1, 2.
Let (X, d) be a complete metric space and T a selfmapping on X satisfying the following condition:
(1.2) d(T x, T y) ≤ a1F1(min{d(x, T x), d(y, T y)}) + a2F2(d(x, T x)d(y, T y)) + a3d(x, y) + a4[d(x, T x) + d(y, T y)] + a5[d(x, T y) + d(y, T x)]
for all x, y in X, where ai= ai(x, y) (i = 1, 2, 3, 4, 5) are nonnegative functions for which there exist three constants K > 0 and λ1, λ2 ∈ (0, 1), such that
the following inequalities:
a1(x, y), a1(x, y) ≤ K, (1.3)
a4(x, y) + a5(x, y) ≤ λ1, (1.4)
a3(x, y) + 2a5(x, y) ≤ λ2, (1.5)
are satisfied for all x, y in X.
If T is asymptotically regular at some x0 in X, then T has a unique fixed point in X and at this point T is continuous.
In this paper, we introduce a general contractive condition similar to the condition (1.2). More precisely, we replace the two functions F1 and F2 by a single function F of two variables (see condition (2.1) below). The aim of this paper is to study the well posedness of the fixed point problem of asymptotically regular mappings satisfying the condition (2.1). We prove (see Theorem 2.1 below) that the fixed point problem for these mappings is well- posed. This work is inspired by the paper [6]. Our main result extends and unifies the results of Ćirić [6], Sharma and Yuel [19], and Guay and Singh [7].
The main result of this paper is Theorem 2.1. It is established in Sec- tion 2. In Section 3, we have gathered some consequences and corollaries.
In Section 4, we have stated two versions of Theorem 2.1 in non complete spaces and in (non complete but) orbitally complete spaces respectively.
2. Main Result
The main result of this paper is the following theorem.
Theorem 2.1. Let R+ be the set of nonnegative reals and let F : R+× R+→ R+ be a function such that F (t, 0) = F (0, t) = 0 and F is continuous at (t, 0) and (0, t) for all t ≥ 0.
Let (X, d) be a complete metric space and T a selfmapping on X satisfying the following condition:
d(T x, T y) ≤ a0F (d(x, T x), d(y, T y)) + a1d(x, y) (2.1)
+ a2[d(x, T x) + d(y, T y)] + a3[d(x, T y) + d(y, T x)]
for all x, y in X, where ai= ai(x, y) (i = 0, 1, 2, 3) are nonnegative functions for which there exist three constants K > 0 and λ1, λ2 ∈ (0, 1), such that
the following inequalities:
a0(x, y) ≤ K, (2.2)
a2(x, y) + a3(x, y) ≤ λ1, (2.3)
a1(x, y) + 2a3(x, y) ≤ λ2, (2.4)
are satisfied for all x, y in X.
If T is asymptotically regular at some x0in X, then the fixed point problem of T is well-posed. Moreover, T is continuous at its unique fixed point.
Proof. Let x0 be a point of X at which T is asymptotically regular.
Then as in the proof of Theorem 1 of [6], one can show that {xn} is a Cauchy sequence, where xn= Tnx0 and that {xn} converges to a fixed point z ∈ X.
To prove the uniqueness of z, let us suppose that u and v are two fixed points of T . From (2.1), with ai= ai(u, v),
d(u, v) = d(T u, T v)
= a0F (0, 0) + a1d(u, v) + a2· 0 + 2a3d(u, v)
= (a1+ 2a3)d(u, v).
Hence, because of (2.4),
(2.5) (1 − λ2)d(u, v) ≤ 0,
which implies v = u.
Now, let {xn} be any arbitrary sequence satisfying limn→∞d(T xn, xn) = 0. For every nonnegative integer n, we denote
(2.6) dn= d(xn, T xn).
Using the triangle inequality, from (2.1) we have d(xn, xm) ≤ dn+ d(T xn, T xm) + dm
≤ dn+ dm+ a0F (dn, dm) + a1d(xn, xm) + a2(dn+ dm) + a3[d(xn, T xm) + d(xm, T xn)],
where ai= ai(xn, xm) for i = 0, 1, 2, 3.
Using again the triangle inequality, we get
d(xn, xm) ≤ (a1+ 2a3)d(xn, xm) + (1 + a2+ a3)(dn+ dm) + a0F (dn, dm).
Hence, because of (2.3), (2.4) and (2.5), we obtain
(2.7) (1 − λ2)d(xn, xm) ≤ (1 + λ1)(dn+ dm) + KF (dn, dm).
Since limn→∞d(xn, T xn) = 0 and F is continuous at (0, 0), then by taking the limit as m tends to infinity we obtain
(2.8) (1 − λ2) lim
n≥m→∞d(xn, xm) = 0, which implies that {xn} is a Cauchy sequence.
Since X is complete, {xn} is convergent to a limit (say) u in X.
Now we show that u is equal to the unique fixed point z of T . We start by proving that T u = u. To get a contradiction, let us suppose that d(u, T u) > 0.
Then, from (2.1) we have
d(u, T u) ≤ d(u, T xn) + d(T xn, T u)
≤ d(u, xn) + d(xn, T xn) + a0F (dn, d(u, T u)) + a1d(xn, u) + a2[dn+ d(u, T u)]
+ a3[d(xn, T u) + d(u, T xn)], where ai= ai(xn, u) for i = 1, 2, 3.
Using the triangle inequality we get
d(u, T u) ≤ a0F (dn, d(u, T u)) + (a2+ a3)d(u, T u)
+ (1 + a2+ a3)d(xn, T xn) + (1 + a1+ 2a3)d(u, xn).
Therefore, from (2.2), (2.3) and (2.4),
d(u, T u) ≤ KF (dn, d(u, T u)) + λ1d(u, T u)
+ (1 + λ1)d(xn, T xn) + (1 + λ2)d(u, xn).
Taking the limit and using the continuity of F at (0, d(u, T u)), we get d(u, T u) ≤ λ1d(u, T u) < d(u, T u),
a contradiction. Therefore, d(u, T u) = 0 That is T u = u. By uniqueness of z, we must have z = u. We conclude that the fixed point problem of T is well-posed.
To prove that T is continuous at z, suppose that un → z = T z. Then from (2.1),
d(T un, z) = d(T un, T z)
≤ a0F [(d(un, T un), 0)] + a1d(un, z)
+ a2d(un, T un) + a3[d(un, z) + d(T un, z)]
≤ KF [(d(un, T un), 0)] + (a1+ a2+ a3)d(un, z) + (a2+ a3)d(T un, z),
where ai= ai(un, u) for i = 1, 2, 3.
Hence, using (2.3) and (2.4),
(2.9) (1 − λ1)d(T un, u) ≤ KF [(d(un, T un), 0)] + (λ1+ λ2)d(un, z).
By letting n go to infinity and using the fact that F (t, 0) = 0, we obtain (1 − λ1) lim sup
n
d(T un, z) ≤ 0,
which implies that limn→∞T un= z. This completes the proof.
3. Consequences and Applications
We have the following corollaries.
Corollary 3.1. Let R+be the set of nonnegative reals and let Fi: R+→ R+ be functions such that Fi(0) = 0 and Fi is continuous at 0 for (i = 1, 2).
Let (X, d) be a complete metric space and T a selfmapping on X satisfying the following condition:
(3.1) d(T x, T y) ≤ b1F1(min{d(x, T x), d(y, T y)}) + b2F2(d(x, T x)d(y, T y)) + b3d(x, y) + b4[d(x, T x) + d(y, T y)] + b5[d(x, T y) + d(y, T x)]
for all x, y in X, where bi= bi(x, y) (i = 1, 2, 3, 4, 5) are nonnegative functions for which there exist three constants K > 0 and λ1, λ2 ∈ (0, 1), such that
the following inequalities:
b1(x, y), b2(x, y) ≤ K, (3.2)
b4(x, y) + b5(x, y) ≤ λ1, (3.3)
b3(x, y) + 2b5(x, y) ≤ λ2, (3.4)
are satisfied for all x, y in X.
If T is asymptotically regular at some x0in X, then the fixed point problem of T is well-posed. Moreover, T is continuous at its unique fixed point.
The proof follows from Theorem 2.1, by considering the functions:
F (s, t) := F1(min{s, t}) + F2(st),
a0(x, y) := max{b1(x, y), b2(x, y)}, and a1:= b3, a2:= b4, a3 := b5. Corollary 3.2. Let α ≥ 0 and β ∈ [0, 1). Let (X, d) be a complete metric space and T a selfmapping on X satisfying the following condition:
(3.5) d(T x, T y) ≤ αmin{d(x, T x), d(y, T y) + d(x, T x)d(y, T y)}
1 + d(x, y) + βd(x, y)
for all x, y in X. If T is asymptotically regular at some x0 in X, then the fixed point problem of T is well-posed. Moreover, T is continuous at its unique fixed point.
The proof follows from Theorem 2.1 by considering the functions F (s, t) := α[(min{s, t}) + st],
a0(x, y) := 1
1 + d(x, y), and a1:= β, a2:= 0, a3:= 0.
Beside these considerations, we can take K = 1, λ1 = λ2:= β.
We observe that the contractive condition (3.5) is more general than the one considered by Sharma and Yuel in [19].
Corollary 3.3. Let (X, d) be a complete metric space and T a selfmap- ping on X satisfying the following condition:
(3.6) d(T x, T y) ≤ pd(x, y) + q[d(x, T x) + d(y, T y)] + r[d(x, T y) + d(y, T x)], where p, q, and r are fixed (nonegative) real numbers such that q + r < 1 and p + 2r < 1.
If T is asymptotically regular at some x0in X, then the fixed point problem of T is well-posed. Moreover, T is continuous at its unique fixed point.
The proof follows from Theorem 2.1, by considering the functions:
F (s, t) := 0, a0(x, y) := 0, and a1 := p, a2 := q, a3 := r.
Beside these considerations, we can take K = 0, λ1 = q + r and λ2:= p + 2r.
Condition (3.6) is the contractive condition, introduced and considered by Guay and Singh in [7].
4. General results in non complete metric spaces
In the case where the metric space (X, d) is not complete, we have the following general result.
Theorem 4.1. Let R+ be the set of nonnegative reals and let F : R+× R+→ R+ be a function such that F (t, 0) = F (0, t) = 0 and F is continuous at (t, 0) and (0, t) for all t ≥ 0. Let (X, d) be a metric space and T a selfmapping on X satisfying condition (2.1) for all x, y in X, where ai = ai(x, y) (i = 0, 1, 2, 3) are nonnegative functions for which there exist three constants K > 0 and λ1, λ2 ∈ (0, 1), such that (2.2)–(2.4) are satisfied for all x, y in X.
If T is asymptotically regular at some x0 in X and the sequence of iter- ates {Tnx0} has a subsequence converging to a point z in X, then z is the the unique fixed point of T and the fixed point problem of T is well-posed.
Moreover, T is continuous at z.
Proof. As in the proof of Theorem 2.1 (see also the proof of Theo- rem 1 in [6]), the sequence of iterates {Tnx0} is a Cauchy sequence. Since it contains a subsequence which converges to the point z, we conclude that limn→∞Tnx0 = z. By using the contractive condition (2.1), one can prove that z is the unique fixed point of T . The rest of the result is obtained by the method of proof as in Theorem 2.1. This completes the proof. In 1974 Ćirić ([4]) has first introduced orbitally complete metric spaces.
Definition 4.1. Let T be a self mapping of a metric space (X, d). If for all x in X every Cauchy sequence of the orbit OX(T ) = {x, T x, T2x, . . .} is convergent in X, then the metric space (X, d) is said T -orbitally complete.
Remark 1. Every complete metric space is T -orbitally complete for any T : X → X. An orbitally complete space may not be complete metric space (Example 1 [21]).
For (non complete but) orbitally complete metric spaces, we have the following result.
Theorem 4.2. Let R+ be the set of nonnegative reals and let F : R+× R+→ R+ be a function such that F (t, 0) = F (0, t) = 0 and F is continuous at (t, 0) and (0, t) for all t ≥ 0. Let (X, d) be a metric space and T a selfmapping on X satisfying condition (2.1) for all x, y in X, where ai = ai(x, y) (i = 0, 1, 2, 3) are nonnegative functions for which there exist three constants K > 0 and λ1, λ2 ∈ (0, 1), such that (2.2)–(2.4) are satisfied for all x, y in X.
If T is asymptotically regular at some x0 in X and (X, d) is T -orbitally complete then the fixed point problem of T is well-posed. Moreover, T is continuous at its unique fixed point.
Proof. As in the proof of Theorem 2.1 (see also the proof of Theorem 1 in [6]), the sequence of iterates {Tnx0} is a Cauchy sequence. Since (X, d) is T -orbitally complete then this sequence converges to a point z in X. By using the contractive condition (2.1), one can prove that z is the unique fixed point of T . The rest of the result is obtained by the method of proof as in
Theorem 2.1. This completes the proof.
Acknowledgement. The author thanks very much the referee for his (her) many valuable comments and useful suggestions.
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Faculté des Sciences-Semlalia Département de Mathématiques Université Cadi Ayyad
Av. Prince My Abdellah, BP. 2390 Marrakech, Maroc, Morocco e-mail: akkouchimo@yahoo.fr
